I don't want to see good homework problems go to waste, so I offer them here in this blog. Twelve new homework problems. Free. They offer a way to learn a bit of elasticity theory. Enjoy.

Problem 1Consider the rod in Fig. 1.20; the x-axis is along the rod’s length and x=0 is where the rod meets the wall. Let the displacementu(x,y) be the change in position of each point in the material in response to the force F. Express the displacement as u_{x}=Ax, u_{y}=0, and u_{z}=0, where A is a constant.

(a) Calculate the normal strain ε_{n}using Eq. 1.24 and Fig. 1.20.

(b) Calculate ε_{n}using the definition ε_{n}=∂u_{x}/∂x . Is it the same as in (a)?

Problem 2Consider the rod in Fig. 1.23; x is horizontal, y is vertical, and y=0 is where the rod meets the floor. Express the displacement as u_{x}=By, u_{y}=0, and u_{z}=0, where B is a constant.

(a) Calculate the shear strainεusing Eq. 1.27 and Fig. 1.23 (assume B_{s}«1)

(b) Calculateusing the definitionε_{s}ε_{s}=∂u_{x}/∂y+∂u_{y}/∂x. Is it the same as in (a)?

Problem 3The normal and shear strains can be combined into a strain tensor, which a 3 x 3 matrix. Define the diagonal components of this tensor (ε_{xx}, ε_{yy}, ε_{zz}) as the normal strain in each direction, and the off-diagonal components (ε_{xy}, ε_{yz}, ε_{zx}) as one half of the shear strain in each direction. For example, ε_{xy}=(∂u_{x}/∂y+∂u_{y}/∂x)/2.

(a) Derive expressions relating each component of the strain tensor to the displacement.

(b) Show that the strain tensor is symmetric (e.g.,ε=_{xy}ε)._{yx}

Note: This expression for the strain tensor is correct for small strains. For large strains it is a more complicated nonlinear function of the displacement (Fung, 1993).

Problem 4The dilatation is defined as the change in volume over the original volume, ΔV/V. For small strains, the dilatation is ε_{xx}+ε_{yy}+ε_{zz}.

(a) Calculate the dilatation for the displacement in Problem 1. Does the volume change?

(b) Calculate the dilatation for the displacement in Problem 2. Does the volume change?

(c) Calculate the dilatation for the displacement u_{x}=Cx, u_{y}=Cy, u_{z}=Cz, where C is a constant. Does the volume change?

(d) Show that the dilatation is equal to the trace of the strain tensor (the trace of a matrix is the sum of the diagonal components) and also is equal to the divergence of the displacement (the divergence is defined in Chapter 4).

Problem 5Consider the displacement u_{x}=Dx, u_{y}=-Dy, u_{z}=0, where D is a constant.

(a) Sketch a plot of the displacement distribution by drawing the displacement vectors over a 5 x 5 grid centered at the origin.

(b) Sketch how a small square in the x-y plane centered at the origin is deformed.

(c) Calculate the strain tensor (defined in Problem 3) for this displacement.

(d) Calculate the dilatation (defined in Problem 4) for this displacement.

Problem 6Repeat the analysis of Problem 5 for the displacement u_{x}=Fy, u_{y}=Fx, u_{z}=0, where F is a constant.

Problem 7Repeat the analysis of Problem 5 for the displacementu._{x}=Hy, u_{y}=-Hx, u_{z}=0, where H is a constantThis is a special case of rigid body motion. Interpret this displacement physically.

Problem 8Like the strain, the stress can be written as a 3 x 3 symmetric tensor. For an isotropic material, the relationship between the components of the stress tensor, s_{ij}, and the strain tensor, ε_{ij}, is s_{ij}=λδ_{ij}(ε_{xx}+ε_{yy}+ε_{zz})+2με_{ij}, where λ and μ are the Lame parameters, andδis the Kronecker delta (1 if i=j, and 0 otherwise)._{ij}

(a) Show that for the case in Problem 1, this relationship reduces to Eq. 1.25 where s_{n}=s_{xx}and ε_{n}=ε. Express the Young’s modulus E in terms of the Lame parameters._{xx}

(b) Show that for the case in Problem 2, this relationship reduces to Eq. 1.28 where s_{s}=sand ε_{xy}_{s}=2ε. Express the shear modulus G in terms of the Lame parameters._{xy}

(c) Show that for the case in Problem 4c, this relationship reduces to Eq. 1.32 where the diagonal components of the stress tensor are given in terms of the pressure p as s=s_{xx}_{yy}=s_{zz}=-p. Express the compressibility κ in terms of the Lame parameters.

Problem 9Figure 1.25 shows that pressure p will exert a net force on an element of fluid only if p is not uniform. Similarly, a stress will exert a net force on an element of tissue only if the stress is not uniform. The equations of mechanical equilibrium (zero net force) are ∂s_{ix}/∂x+∂s_{iy}/∂y+∂s_{iz}/∂z=0, where i is either x, y, or z.

(a) Substitute the relationship from Problem 8 into these equations, and derive three equations of mechanical equilibrium written in terms of the strain tensor.

(b) Substitute the relationships between the components of the strain tensor and the displacement found in Problem 3 and derive the equations of mechanical equilibrium in terms of the displacement.

Problem 10Figure 1.20, showing a rod subject to a force along its length, is a simplification. Actually, the cross-sectional area of the rod shrinks as the rod lengthens. A better representation of the displacement than that given in Problem 1 would be u_{x}=Ax, u_{y}=-Aνy, and u_{z}=-Aνz, where A is a constant and ν is the Poisson’s ratio.

(a) Use the results of Problem 4 to calculate the dilatation.

(b) What value of Poisson’s ratio corresponds to an incompressible material (zero dilatation)?

(c) For an isotropic material, -1 « ν « 0.5. How would a material with negative ν behave?

Elliott et al. (2002) measured Poisson’s ratio for articular (joint) cartilage under tension and found 1«ν«2. This large value is possible because cartilage is anisotropic: its properties depend on direction.

Problem 11Many biological tissues are composed mainly of water and are therefore nearly incompressible. To analyze such a tissue, start with the stress-strain relationship in Problem 8. (Assumeuall derivatives in the z direction are zero; the case of “plane strain”.)_{z}=0 and

(a) For an incompressible tissue, ε_{xx}+ε_{yy}goes to zero and λ goes to infinity such that λ(ε_{xx}+ε_{yy}) is finite. Set it equal to –p, where p is the pressure.

(b) The displacement can be found from a stream function φ(x,y), where u_{x}=∂φ/∂y and u_{y}=-∂φ/∂x. Show that these definitions ensure that the dilatation is zero. Express the strain tensor in terms of φ.

(c) Write the components of the stress tensor (s_{xx}, s_{yy}, s_{xy}) in terms of p and φ.

(d) Use the analysis of Problem 9 to derive the equations of mechanical equilibrium in terms of p and φ.

(e) Manipulate these equations to find two new equations, one for p only and one for φ only. (Hint: try taking derivatives of the equations).

Problem 12Start with the stress-strain relationship in Problem 8 and modify it to describe a two-dimensional sheet of cardiac muscle (Ohayon and Chadwick 1988). (Assumeall derivatives in the z direction are zero; the case of “plane strain”.)u_{z}=0 and

(a) Cardiac tissue is nearly incompressible. For an incompressible tissue,εgoes to zero and λ goes to infinity such that λ(_{xx}+ε_{yy}ε) is finite. Set it equal to –p, where p is the pressure._{xx}+ε_{yy}

(b) Cardiac muscle can develop an active tension T along the myocardial fibers caused by the interaction of actin and myosin molecules. Assume the fibers lie along the x direction, and add the term T to the expression fors._{xx}

(c) The extracellular space consists of collagen fibers that can exert a shear force. Assume the collagen is isotropic, and interpret μ in Problem 8 as the collagen’s shear modulus.

(d) Derive expressions for,s_{xx}, ands_{yy}in terms of p, μ, T, and the strain tensor.s_{xy}

(e) Assume a solution u_{x}=-Ax, u_{y}=Ay, and p=P, where A and P are constants. If the tissue is free at its edges, then it must have zero stress throughout. Use this condition to derive expressions for A and P in terms of T and μ.

(f) Let T=3 x 10^{4}Pa and μ=10^{4}Pa (typical for cardiac tissue). Calculate values for A and P. Are the strains small? Sketch qualitatively the displacement distribution.

Ohayon, J. and R. S. Chadwick (1988) Effects of collagen microstructure on the mechanics of the left ventricle. Biophys. J. 54:1077-1088.

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